Finite Group Databasebeta

← #13107119

Subgroups — #13107119

Order 1600.

1712 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 1712 conjugacy classes of subgroups — too many to draw. Only its 259 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C5⋊D5)C212yesmaximal, complemented
2C2×D4×C5⋊D5C212yesmaximal, complemented
3C52×#120C211yesmaximal, characteristic, complemented
4#11717260C211yesmaximal, characteristic, complemented
5#11717259C212yesmaximal, complemented
6#11717257C211yesmaximal, characteristic, complemented
7#11717204C212yesmaximal, complemented
8#11717202C212yesmaximal, complemented
9#11717201C212yesmaximal, complemented
10C24×C52C2211yescharacteristic, complemented
11C23×C5⋊D5C2212yescomplemented
12C23×C5⋊D5C2211yescharacteristic, complemented
13C2×C52×D4C2212yescomplemented
14C2×C52×D4C2211yescharacteristic, complemented
15C2×C22⋊(C5⋊D5)—28no—
16C2×C22⋊(C5⋊D5)C2212yes—
17C2×C22⋊(C5⋊D5)C2212yescomplemented
18C2×C22⋊(C5⋊D5)C2212yescomplemented
19C2×C22⋊(C5⋊D5)C2212yescomplemented
20C2×C22⋊(C5⋊D5)C2212yescomplemented
21C22×#1039C2212yescomplemented
22D4×C5⋊D5—28no—
23C2×C4⋊(C5⋊D5)C2212yescomplemented
24C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
25C2×C4×C5⋊D5C2212yescomplemented
26C52×C22⋊C4C2212yescomplemented
27C52×C22⋊C4C2211yescharacteristic, complemented
28C22⋊#1039C2211yescharacteristic, complemented
29#87808C2212yescomplemented
30#87808C2212yescomplemented
31#87806C2212yescomplemented
32C4×#1039C2211yescharacteristic, complemented
33#65924—56nomaximal
34C23×C52—24no—
35C23×C52—41no—
36C23×C52C2311yescharacteristic
37C23×C52C2311yescharacteristic
38C23×C52C2312yes—
39C22×C5⋊D5—24no—
40C22×C5⋊D5—24no—
41C22×C5⋊D5—24no—
42C22×C5⋊D5—41no—
43C22×C5⋊D5—42no—
44C22×C5⋊D5C2312yes—
45C22×C5⋊D5C2312yes—
46C52×D4—24no—
47C52×D4—41no—
48C2×C4×C52C2311yescharacteristic
49C2×C4×C52C2312yes—
50C22⋊(C5⋊D5)—24no—
51C22⋊(C5⋊D5)—24no—
52C22⋊(C5⋊D5)—24no—
53C22⋊(C5⋊D5)—24no—
54C22⋊(C5⋊D5)D418yescomplemented
55C2×#1039—24no—
56C2×#1039C2312yes—
57C2×#1039C2312yes—
58C4⋊(C5⋊D5)—24no—
59C4⋊(C5⋊D5)—41no—
60C4×C5⋊D5—24no—
61C22×C22⋊D5—512no—
62C2×D4×D5—512no—
63C5×#120D516yescomplemented
64#4086—56no—
65#4085—512no—
66#4083—56no—
67#4030—512no—
68#4028—512no—
69#4027—512no—
70C22×C52—22no—
71C22×C52—22no—
72C22×C52—22no—
73C22×C52—22no—
74C22×C52—21no—
75C22×C52—21no—
76C22×C52—44no—
77C22×C52C2×D414yescomplemented
78C22×C52C2411yescharacteristic
79C2×C5⋊D5—22no—
80C2×C5⋊D5—22no—
81C2×C5⋊D5—22no—
82C2×C5⋊D5—22no—
83C2×C5⋊D5—22no—
84C2×C5⋊D5—44no—
85C2×C5⋊D5—48no—
86C2×C5⋊D5C2×D414yescomplemented
87C4×C52—22no—
88C4×C52—21no—
89#1039—22no—
90#1039C2×D414yescomplemented
91C24×C5C2×D516yescomplemented
92C23×D5—56no—
93C23×D5—512no—
94C2×C5×D4C2×D5112yescomplemented
95C2×C5×D4C2×D516yescomplemented
96C2×C22⋊D5—512no—
97C2×C22⋊D5—512no—
98C2×C22⋊D5—512no—
99C2×C22⋊D5—512no—
100C2×C22⋊D5—512no—
101C2×C22⋊D5—1048no—
102C22×Dic5—512no—
103D4×D5—1048no—
104C2×D20—512no—
105C2×D20—56no—
106C2×C4×D5—512no—
107C5×C22⋊C4C2×D5112yescomplemented
108C5×C22⋊C4C2×D516yescomplemented
109C22⋊Dic5—56no—
110#685—512no—
111#685—512no—
112#683—512no—
113C4×Dic5—56no—
114D42—251noSylow
115C2×C52—24no—
116C2×C52—41no—
117C2×C52—41no—
118C2×C52C22×D412yes—
119C2×C52H(5,𝔽2)11yescharacteristic
120C5⋊D5—24no—
121C5⋊D5—42no—
122C23×C5—224no—
123C23×C5—46no—
124C23×C5C22×D516yes—
125C23×C5C22×D516yes—
126C23×C5C22×D5112yes—
127C22×D5—512no—
128C22×D5—512no—
129C22×D5—1024no—
130C22×D5—1024no—
131C22×D5—1024no—
132C22×D5—206no—
133C22×D5—2012no—
134C5×D4—224no—
135C5×D4—46no—
136C2×C4×C5C22×D516yes—
137C2×C4×C5C22×D5112yes—
138C22⋊D5—548no—
139C22⋊D5—1024no—
140C22⋊D5—1024no—
141C22⋊D5—1024no—
142C22⋊D5—1024no—
143C2×Dic5—512no—
144C2×Dic5—512no—
145C2×Dic5—1024no—
146D20—1024no—
147D20—206no—
148C4×D5—1024no—
149C22×D4—252no—
150C22×D4—252no—
151#127—251no—
152#121—252no—
153#121—252no—
154#120—251no—
155#120—252no—
156#120C5⋊D511yescharacteristic, complemented
157C4×D4—252no—
158C52D4211yescharacteristic, complemented, Sylow
159C22×C5—212no—
160C22×C5—212no—
161C22×C5—212no—
162C22×C5—212no—
163C22×C5—26no—
164C22×C5—26no—
165C22×C5—424no—
166C22×C5D4×D5124yescomplemented
167C22×C5C23×D516yes—
168C2×D5—524no—
169C2×D5—1012no—
170C2×D5—1012no—
171C2×D5—1012no—
172C2×D5—1012no—
173C2×D5—1012no—
174C2×D5—2024no—
175C2×D5—2048no—
176C4×C5—212no—
177C4×C5—26no—
178Dic5—524no—
179Dic5—1012no—
180C24—251no—
181C24—252no—
182C24C2×C5⋊D511yescharacteristic, complemented
183C2×D4—252no—
184C2×D4—252no—
185C2×D4—252no—
186C2×D4—252no—
187C2×D4—252no—
188C2×D4—252no—
189C2×D4—251no—
190C2×D4—508no—
191C2×D4—508no—
192C2×D4C2×C5⋊D512yescomplemented
193C2×D4C2×C5⋊D511yescharacteristic, complemented
194C22×C4—252no—
195C22×C4—252no—
196C4⋊C4—252no—
197C22⋊C4—251no—
198C22⋊C4—252no—
199C22⋊C4—252no—
200C22⋊C4C2×C5⋊D512yescomplemented
201C22⋊C4C2×C5⋊D511yescharacteristic, complemented
202C42—251no—
203C2×C5—224no—
204C2×C5—46no—
205C2×C5—46no—
206C2×C5C2×D4×D5112yes—
207C2×C5#414416yes—
208D5—1024no—
209D5—2012no—
210C23—24no—
211C23—41no—
212C23—252no—
213C23—252no—
214C23—504no—
215C23—504no—
216C23—504no—
217C23—1002no—
218C23—1001no—
219C23C22×C5⋊D511yescharacteristic
220C23C22×C5⋊D511yescharacteristic
221C23C22×C5⋊D512yes—
222D4—24no—
223D4—41no—
224D4—258no—
225D4—504no—
226D4—504no—
227D4—504no—
228D4—504no—
229D4—504no—
230D4—1001no—
231C2×C4—252no—
232C2×C4—252no—
233C2×C4—504no—
234C2×C4—504no—
235C2×C4C22×C5⋊D511yescharacteristic
236C2×C4C22×C5⋊D512yes—
237C5#6592416yescomplemented
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—21no—
243C22—21no—
244C22—44no—
245C22—254no—
246C22—502no—
247C22—502no—
248C22—502no—
249C22—502no—
250C22—502no—
251C22—1008no—
252C22—1004no—
253C22D4×C5⋊D514yescomplemented
254C22C23×C5⋊D511yescharacteristic
255C4—22no—
256C4—21no—
257C4—254no—
258C4—502no—
259C2—24no—
260C2—41no—
261C2—41no—
262C2—504no—
263C2—1002no—
264C2C2×D4×C5⋊D512yes—
265C2#1171751711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#120C2
hypercenter#120C5⋊D5
hypocenterC52D42
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C22⋊(C5⋊D5)C2semidirect
2. C2×D4×C5⋊D5C2semidirect
3. C52×#120C2semidirect
4. #11717260C2semidirect
5. #11717259C2semidirect
6. #11717257C2semidirect
7. #11717204C2semidirect
8. #11717202C2semidirect
9. #11717201C2semidirect
10. C24×C52C22semidirect
11. C23×C5⋊D5C22semidirect
12. C23×C5⋊D5C22semidirect
13. C2×C52×D4C22semidirect
14. C2×C52×D4C22semidirect
17. C2×C22⋊(C5⋊D5)C22semidirect
18. C2×C22⋊(C5⋊D5)C22semidirect
19. C2×C22⋊(C5⋊D5)C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
21. C22×#1039C22semidirect
23. C2×C4⋊(C5⋊D5)C22semidirect
24. C2×C4⋊(C5⋊D5)C22semidirect
25. C2×C4×C5⋊D5C22semidirect
26. C52×C22⋊C4C22semidirect
27. C52×C22⋊C4C22semidirect
28. C22⋊#1039C22semidirect
29. #87808C22semidirect
30. #87808C22semidirect
31. #87806C22semidirect
32. C4×#1039C22semidirect
54. C22⋊(C5⋊D5)D4semidirect
63. C5×#120D5semidirect
77. C22×C52C2×D4semidirect
86. C2×C5⋊D5C2×D4semidirect
90. #1039C2×D4semidirect
91. C24×C5C2×D5semidirect
94. C2×C5×D4C2×D5semidirect
95. C2×C5×D4C2×D5semidirect
107. C5×C22⋊C4C2×D5semidirect
108. C5×C22⋊C4C2×D5semidirect
156. #120C5⋊D5semidirect
158. C52D42semidirect
166. C22×C5D4×D5semidirect
182. C24C2×C5⋊D5semidirect
192. C2×D4C2×C5⋊D5semidirect
193. C2×D4C2×C5⋊D5semidirect
200. C22⋊C4C2×C5⋊D5semidirect
201. C22⋊C4C2×C5⋊D5semidirect
237. C5#65924semidirect
253. C22D4×C5⋊D5semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C22⋊(C5⋊D5)2
C2×D4×C5⋊D52
C52×#1202
#117172602
#117172592
#117172572
#117172042
#117172022
#117172012
#659245